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CN110865617B - Equipment opportunity maintenance and production scheduling integrated optimization method under time-varying working condition - Google Patents

Equipment opportunity maintenance and production scheduling integrated optimization method under time-varying working conditionDownload PDF

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CN110865617B
CN110865617BCN201911111516.2ACN201911111516ACN110865617BCN 110865617 BCN110865617 BCN 110865617BCN 201911111516 ACN201911111516 ACN 201911111516ACN 110865617 BCN110865617 BCN 110865617B
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肖雷
汤俊萱
鲍劲松
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Donghua University
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Abstract

Maintenance of some equipment in engineering practice can only restore it to a degraded state and not repair it as new. Furthermore, in mass production systems, production processing tasks are often not interrupted, and therefore maintenance activities can only be moved to be performed before or after a processing task. In addition, the influence on the reliability degradation of the equipment is different in consideration of different working conditions of different production and processing tasks. Aiming at the problems, the invention provides a method for integrating and optimizing equipment opportunity maintenance and production scheduling under the time-varying working condition. The invention considers the equipment maintenance and the production scheduling integrated optimization under different production systems and different maintenance effects, ensures the reliability of the equipment and improves the production efficiency of the production system.

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Translated fromChinese
时变工况下设备机会维护和生产调度集成优化方法Integrated optimization method of equipment opportunity maintenance and production scheduling under time-varying conditions

技术领域technical field

本发明涉及设备维护和生产调度集成优化技术。具体地说是在考虑不同加工任务的操作工况的情况下，结合批量生产系统的特点，建立在批量生产系统中设备的机会非完美性预防性维护和生产调度集成优化模型。The invention relates to an integrated optimization technology of equipment maintenance and production scheduling. Specifically, considering the operating conditions of different processing tasks and combining the characteristics of batch production systems, an integrated optimization model of opportunistic imperfection preventive maintenance and production scheduling of equipment in batch production systems is established.

背景技术Background technique

在生产实践中，一个产品往往需要在多台不同的设备上进行加工才能完成其生产任务。在此系统中的任意一台设备的失效或维护都会对生产系统的生产造成影响，因此有必要将设备的维护和生产调度进行集成优化。在进行优化时，需要考虑维护的策略、维护效力、维护时间和维护成本等多方面的影响。此外，不同的生产模式要求设备维护方式也不同，因此，在对设备进行维护和生产调度集成优化时，不仅需要考虑设备的当前状态，还需要考虑设备维护后的状态以及执行维护后设备的退化轨迹，还要考虑时间因素(维护时间、加工时间、设备启动和停机时间、作业转换时间)、成本因素(维护成本、加工成本、交货延迟成本)、生产任务的自身特点以及生产模式和生产系统特性。In production practice, a product often needs to be processed on multiple different equipments to complete its production tasks. The failure or maintenance of any piece of equipment in this system will affect the production of the production system, so it is necessary to integrate and optimize equipment maintenance and production scheduling. When optimizing, it is necessary to consider the influence of maintenance strategy, maintenance effectiveness, maintenance time and maintenance cost. In addition, different production modes require different equipment maintenance methods. Therefore, when performing integrated optimization of equipment maintenance and production scheduling, not only the current state of the equipment, but also the state of the equipment after maintenance and the degradation of the equipment after performing maintenance must be considered. trajectory, but also consider time factors (maintenance time, processing time, equipment start-up and downtime, job conversion time), cost factors (maintenance costs, processing costs, delivery delay costs), the characteristics of production tasks, and the production mode and production system characteristics.

发明内容Contents of the invention

本发明的目的是：建立在批量生产系统中设备的机会非完美性预防维护和生产调度集成优化模型。The purpose of the present invention is to establish an integrated optimization model of chance imperfection preventive maintenance and production scheduling of equipment in a batch production system.

为了达到上述目的，本发明的技术方案是提供了一种时变工况下设备机会维护和生产调度集成优化方法，其特征在于，包括以下步骤：In order to achieve the above object, the technical solution of the present invention is to provide an integrated optimization method for equipment opportunity maintenance and production scheduling under time-varying working conditions, which is characterized in that it includes the following steps:

第一步、确定问题和假设Step 1. Identify the Question and Hypothesis

假设一个工作间job-shop含有M个设备，一个包括N个批量的加工任务要在此工作间job-shop上进行处理，在加工任务开始前所有的批量都已准备就绪，N个批量之间是相互独立的；对于每一个批量有一个交货截止时间，如果批量的完成时间滞后于相应的交货截止时间，则会产生延迟成本作为成本；设备为转换不同批量的启动时间及批量间的转移时间忽略不计；如果设备发生故障，则对设备进行小修，使其恢复到可以运行的状态但是不改变其风险率，设备的小修时间忽略不计；假设设备上进行成批生产模式，但是预防性维护PM行为不能打断一个正在进行加工的设备，因此预防性维护PM行为不得不提前至加工任务前，或推迟至加工任务后；Assume that a job-shop contains M equipment, and a processing task including N batches needs to be processed in this job-shop. Before the processing task starts, all batches are ready. Between N batches are independent of each other; there is a delivery cut-off time for each batch, and if the completion time of the batch lags behind the corresponding delivery cut-off time, delay costs will be generated as costs; The transfer time is negligible; if the equipment fails, minor repairs are made to the equipment to restore it to an operational state without changing its risk rate, and the minor repair time of the equipment is negligible; it is assumed that batch production mode is performed on the equipment, but prevent The maintenance PM behavior cannot interrupt a device that is being processed, so the preventive maintenance PM behavior has to be advanced before the processing task, or postponed until after the processing task;

第二步、确定优化目标The second step is to determine the optimization goal

集成优化的目标是通过确定工作间job-shop上各个设备上的加工顺序和预防性维护PM的执行时间使总成本最小，总成本包括延迟成本、预防性维护PM成本和期望小修成本，一个批量的延迟成本与它的完成时间有关，预防性维护PM成本和小修成本与预防性维护PM的执行次数和时间有关，目标函数为：The goal of integrated optimization is to minimize the total cost by determining the processing sequence on each device on the job-shop and the execution time of preventive maintenance PM. The total cost includes delay cost, preventive maintenance PM cost and expected minor repair cost, a batch The delay cost of is related to its completion time, the preventive maintenance PM cost and minor repair cost are related to the execution times and time of preventive maintenance PM, and the objective function is:

min Ctotal＝CT+CP+CFmin Ctotal ＝CT +CP +CF

式中，Ctotal为总成本；In the formula, Ctotal is the total cost;

CT为总延迟成本，

Figure GDA0003719825160000021
N为批量总数，Tn为批量n的延迟成本，
Figure GDA0003719825160000022
Ln为批量n的延迟，
Figure GDA0003719825160000023
Em,n为批量n在设备m上的完成时间，Dn为给定的批量n的预计完成时间，
Figure GDA0003719825160000024
为批量n的单位时间内延迟成本；CT is the total delay cost,
Figure GDA0003719825160000021
N is the total number of batches, Tn is the delay cost of batch n,
Figure GDA0003719825160000022
Ln is the delay of batch n,
Figure GDA0003719825160000023
Em,n is the completion time of batch n on device m, Dn is the estimated completion time of a given batch n,
Figure GDA0003719825160000024
Delay cost per unit time for batch n;

CP为总预防性维护PM成本：CP is the total preventive maintenance PM cost:

Figure GDA0003719825160000025
Im为任务周期内设备m上执行预防性维护PM的总次数，
Figure GDA0003719825160000026
为一次预防性维护PM成本；
Figure GDA0003719825160000025
Im is the total number of preventive maintenance PMs performed on equipment m in the task period,
Figure GDA0003719825160000026
PM cost for one preventive maintenance;

CF为总期望小修成本：CF is the total expected minor repair cost:

Figure GDA0003719825160000027
Figure GDA0003719825160000028
为设备m上一次小修成本，
Figure GDA0003719825160000029
为在基准工况下设备m在第i个预防性维护PM周期内的等价加工时间，
Figure GDA00037198251600000210
为在基准工况下设备m进行完最后一次预防性维护PM后的等价加工时间，
Figure GDA00037198251600000211
为第i个预防性维护PM周期内设备m的基准失效率函数；
Figure GDA0003719825160000027
Figure GDA0003719825160000028
is the last minor repair cost of equipment m,
Figure GDA0003719825160000029
is the equivalent processing time of equipment m in the i-th preventive maintenance PM cycle under the baseline condition,
Figure GDA00037198251600000210
is the equivalent processing time after the last preventive maintenance PM of equipment m under the baseline condition,
Figure GDA00037198251600000211
is the baseline failure rate function of equipment m in the i-th PM period of preventive maintenance;

第三步、确定批量完成时间The third step is to determine the batch completion time

一台设备上一个批量的完成时间与四项因素有关：(1)此台设备上上一批量的完成时间；(2)此台设备上此批量的加工时间；(3)此批量在上一台设备上的完成时间；(4)如果在加工此批量前需要进行预防性维护PM，则维护的执行时间；The completion time of a batch on a piece of equipment is related to four factors: (1) the completion time of the previous batch on this equipment; (2) the processing time of this batch on this equipment; (4) If preventive maintenance PM is required before processing this batch, the execution time of maintenance;

第四步、确定预防性维护PM周期内的设备可靠性Step 4. Determine equipment reliability during the PM cycle of preventive maintenance

确定初始执行预防性维护PM时间的成本率函数如下式：The cost rate function to determine the initial PM time for preventive maintenance is as follows:

Figure GDA0003719825160000031
Figure GDA0003719825160000031

式中，

Figure GDA0003719825160000032
为设备m在第i个预防性维护PM周期内的维护成本率，
Figure GDA0003719825160000033
为设备m上的执行一次预防性维护PM所需的时间；In the formula,
Figure GDA0003719825160000032
is the maintenance cost rate of equipment m in the i-th PM cycle of preventive maintenance,
Figure GDA0003719825160000033
is the time required to perform a preventive maintenance PM on equipment m;

假设在设备m的第i个周期内已经有

Figure GDA0003719825160000034
个加工完或正在加工的批量，在该周期内设备的风险率函数写为：Assume that during the i-th cycle of device m there has been
Figure GDA0003719825160000034
A batch that has been processed or is being processed, the risk rate function of the equipment in this period is written as:

Figure GDA0003719825160000035
Figure GDA0003719825160000035

Figure GDA0003719825160000036
Figure GDA0003719825160000036

式中，

Figure GDA0003719825160000037
是0-1变量，如果设备m上的第i个预防性维护PM周期内的第j个加工是批量n，则为1；
Figure GDA0003719825160000038
为在第i个预防性维护PM周期中第j个加工过程中设备m的风险率函数；In the formula,
Figure GDA0003719825160000037
is a 0-1 variable, which is 1 if the j-th process in the i-th preventive maintenance PM cycle on equipment m is batch n;
Figure GDA0003719825160000038
is the risk rate function of equipment m in the j-th process in the i-th preventive maintenance PM cycle;

在第1个预防性维护PM周期内设备在进行第1项加工时的设备可靠性为：During the first PM cycle of preventive maintenance, the equipment reliability when the equipment is performing the first processing is:

Figure GDA0003719825160000041
Figure GDA0003719825160000041

式中，

Figure GDA0003719825160000042
表示在第1个预防性维护PM周期内设备在进行第1项加工时的可靠性，
Figure GDA0003719825160000043
是在基准工况下设备的等价可靠性，
Figure GDA0003719825160000044
是设备m上第k个加工时的工况调节参数，
Figure GDA0003719825160000045
为在基准工况下在第i个预防性维护PM周期内设备m的风险率函数；In the formula,
Figure GDA0003719825160000042
Indicates the reliability of the equipment during the first item of processing during the first PM cycle of preventive maintenance,
Figure GDA0003719825160000043
is the equivalent reliability of the equipment under the reference condition,
Figure GDA0003719825160000044
is the working condition adjustment parameter of the kth processing on the equipment m,
Figure GDA0003719825160000045
is the risk rate function of equipment m in the i-th preventive maintenance PM cycle under the baseline condition;

在第1个预防性维护PM周期内设备m在进行第1个加工时设备的风险率函数为：In the first PM cycle of preventive maintenance, the risk rate function of equipment m when it is performing the first processing is:

Figure GDA0003719825160000046
Figure GDA0003719825160000046

在第1个预防性维护PM周期内，设备m在进行第2个加工时设备的可靠性为：In the first PM cycle of preventive maintenance, the reliability of the equipment m during the second processing is:

Figure GDA0003719825160000047
Figure GDA0003719825160000047

Figure GDA0003719825160000048
Figure GDA0003719825160000048

其中：in:

Figure GDA0003719825160000049
Figure GDA0003719825160000049

其中，Sm,n,k为如果设备m上第k个加工是批量n，设备m的加工开始时间；Among them, Sm,n,k is the processing start time of equipment m if the kth processing on equipment m is batch n;

设备m在第1个预防性维护PM周期内第2个加工时设备的风险率函数为：The risk rate function of equipment m during the second processing in the first preventive maintenance PM cycle is:

Figure GDA0003719825160000051
Figure GDA0003719825160000051

设备m在第1个预防性维护PM周期内第k次加工时设备m的可靠性为：The reliability of equipment m during the k-th processing in the first preventive maintenance PM cycle is:

Figure GDA0003719825160000052
Figure GDA0003719825160000052

设备m在第1个预防性维护PM周期内第k次加工过程中设备的风险率函数为：The risk rate function of equipment m during the k-th processing in the first preventive maintenance PM cycle is:

Figure GDA0003719825160000053
Figure GDA0003719825160000053

由于前文假设，预防性维护PM为非完美性，引入役龄递减因子和失效率递增因子，设备m在第i个预防性维护PM周期内设备的基准风险率函数为：Due to the previous assumption that the preventive maintenance PM is not perfect, and the service life decreasing factor and the failure rate increasing factor are introduced, the basic risk rate function of the equipment m in the i-th preventive maintenance PM cycle is:

Figure GDA0003719825160000054
Figure GDA0003719825160000054

第五步、使用随机键遗传算法GA进行集成优化。The fifth step is to use the random key genetic algorithm GA for integrated optimization.

优选地，通过计算在每个设备上的第一个加工的完成时间计算在此工作间job-shop中其他批量的完成时间，则有：Preferably, the completion time of other batches in this job-shop is calculated by calculating the completion time of the first processing on each device, then:

若在此工作间job-shop上分配给每台设备上的第一个加工批量互不相同，则设备上的第一个加工的完成时间为：If the first processing batches assigned to each device in this job-shop are different from each other, the completion time of the first processing on the device is:

Figure GDA0003719825160000061
Figure GDA0003719825160000061

Figure GDA0003719825160000062
Figure GDA0003719825160000062

式中，Em,n,1为在设备m上如果第1个加工是批量n它的完成时间；xm,n,1为0-1变量，如果设备m上第1个加工是批量n，则xm,n,1的值为1；pm,n为批量n在设备m上的加工时间；In the formula, Em,n,1 is the completion time if the first processing on equipment m is batch n; xm,n,1 is a 0-1 variable, if the first processing on equipment m is batch n , then the value of xm,n,1 is 1; pm,n is the processing time of batch n on equipment m;

若在此job-shop上分配给每台设备上的第一个加工批量有相同的，则设备上的第一个加工的完成时间为：If the first processing batch assigned to each device on this job-shop has the same number, the completion time of the first processing on the device is:

Figure GDA0003719825160000063
Figure GDA0003719825160000063

Figure GDA0003719825160000064
Figure GDA0003719825160000064

如果在设备m上的第k次加工为批量n，它的完成时间从以下几个方面进行推断：如果该批量已经在其他设备上进行了加工，则它的完成时间；设备m上第k-1个加工的完成时间；如果要执行预防性维护PM，则预防性维护PM的执行时间，计算公式为：If the kth processing on equipment m is batch n, its completion time is inferred from the following aspects: if the batch has been processed on other equipment, its completion time; The completion time of 1 machining; if the preventive maintenance PM is to be executed, the execution time of the preventive maintenance PM, the calculation formula is:

Figure GDA0003719825160000065
Figure GDA0003719825160000065

Figure GDA0003719825160000066
Figure GDA0003719825160000066

式中，

Figure GDA0003719825160000067
为设备m上的执行一次预防性维护PM所需的时间，k与k'相等或不相等，如果k＝k'，说明在设备m和设备m'上批量n所处的顺序是相同的，否则，设备m和设备m'上批量n所处的顺序不相同。In the formula,
Figure GDA0003719825160000067
It is the time required to perform a preventive maintenance PM on equipment m, k and k' are equal or not equal, if k=k', it means that the order of batch n on equipment m and equipment m' is the same, Otherwise, the order of batch n on device m and device m' is not the same.

优选地，所述第五步包括以下步骤：Preferably, the fifth step includes the following steps:

首先，定义染色体类型，染色体组是由两部分构成，第1部分为设备上批量的分配，它是一个排序问题，因此，产生一些随机键并将它们按照大小排序，进而得到了加工顺序；第2部分为预防性维护PM决策，此不为排序问题；First, define the chromosome type. The chromosome group is composed of two parts. The first part is the batch allocation on the device. It is a sorting problem. Therefore, some random keys are generated and sorted according to their size, and then the processing order is obtained; thesecond part Part 2 is PM decision-making for preventive maintenance, which is not a ranking issue;

随后，进行交叉：计算经过编码后每个染色体组的适应度，以目标函数作为适应度函数，为保持随机键遗传算法GA的进化为单调非减性，采用精英策略，在每次迭代过程中把精英记录下来并放回交配池中，仅在加工顺序部分执行交叉操作；在预防性维护PM决策部分不进行交叉操作，最小维修成本率与设备在执行前一个预防性维护PM之后的服役时间有关，预防性维护PM决策矩阵是与加工顺序部分有关，交叉操作仅作用于随机键部分；所有的染色体组和他们的键分为两部分：一部分为精英组，一部分为非精英组，精英组中的染色体组和非精英组中的染色体组进行交配，执行单点交叉，并进入凸集理论加速算法的收敛；在交叉过程中，交叉点随机选择，父代在交叉点左侧保持不变，在右侧的部分彼此交换，使用线性连接方法获得交叉点的数值；Then, crossover: calculate the fitness of each chromosome group after encoding, and use the objective function as the fitness function. In order to keep the evolution of the random key genetic algorithm GA as monotonous and non-reduced, an elite strategy is adopted. In each iteration process Record the elites and put them back in the mating pool, only perform cross operations in the processing sequence part; do not perform cross operations in the preventive maintenance PM decision part, the minimum repair cost rate and the service time of the equipment after the execution of the previous preventive maintenance PM Related, the PM decision matrix for preventive maintenance is related to the processing order part, and the crossover operation only acts on the random key part; all chromosome groups and their keys are divided into two parts: one part is the elite group, the other part is the non-elite group, and the elite group Mate the chromosome group in the non-elite group and the chromosome group in the non-elite group, perform single-point crossover, and enter the convergence of the convex set theory to accelerate the algorithm; during the crossover process, the intersection point is randomly selected, and the parent remains unchanged on the left side of the intersection point , the parts on the right are exchanged with each other, and the value of the intersection point is obtained using the linear connection method;

之后，进行变异：变异发生在加工顺序部分和预防性维护PM决策部分，在一个染色体上随机的选择两个基因，然后执行凸连接变异，与交叉算子相同，变异也仅在随机键上进行，对于变异发生在预防性维护PM决策部分的情况，突变在预防性维护PM决策部分进行，在预防性维护PM决策部分，选择某个非零的位置，因为非零的位置为理论上执行预防性维护PM的时间；在突变位置，突变后的结果为1减去原来的值。After that, mutation: the mutation occurs in the processing order part and the preventive maintenance PM decision part. Two genes are randomly selected on a chromosome, and then the convex connection mutation is performed, which is the same as the crossover operator, and the mutation is only performed on the random key. , for the case where the mutation occurs in the PM decision-making part of the preventive maintenance, the mutation is carried out in the PM decision-making part of the preventive maintenance. The time to maintain the PM; at the mutation position, the result after the mutation is 1 minus the original value.

本发明具有如下优点：The present invention has the following advantages:

1)本方法可以在保持设备具有较高的可靠性的前提下优化设备上批量的加工顺序。1) This method can optimize the batch processing sequence on the equipment on the premise of maintaining high reliability of the equipment.

2)本方法与工程实践中常用的提前和滞后定周期预防性维护策略相比，取得了较好的效果。2) This method achieves better results compared with the early and late fixed-period preventive maintenance strategies commonly used in engineering practice.

附图说明Description of drawings

图1为在一个任务周期内不同批量下的设备可靠性；Figure 1 shows the reliability of equipment under different batches within a task cycle;

图2为在有限时间内批量和PM提前和PM滞后的关系图；Fig. 2 is the relationship diagram of batch and PM advance and PM lag in limited time;

图3为在给定加工顺序的情况下验证本文提出的机会维护；Figure 3 is a verification of the opportunistic maintenance proposed in this paper under the given processing sequence;

图4为集成优化模型的验证。Figure 4 is the verification of the integrated optimization model.

具体实施方式Detailed ways

下面结合具体实施例，进一步阐述本发明。应理解，这些实施例仅用于说明本发明而不用于限制本发明的范围。此外应理解，在阅读了本发明讲授的内容之后，本领域技术人员可以对本发明作各种改动或修改，这些等价形式同样落于本申请所附权利要求书所限定的范围。Below in conjunction with specific embodiment, further illustrate the present invention. It should be understood that these examples are only used to illustrate the present invention and are not intended to limit the scope of the present invention. In addition, it should be understood that after reading the teachings of the present invention, those skilled in the art can make various changes or modifications to the present invention, and these equivalent forms also fall within the scope defined by the appended claims of the present application.

本发明方法部分1：确定问题和假设Invention Method Part 1: Determining the Problem and Hypothesis

假设一个job-shop含有M个设备。一个包括N个批量的加工任务要在此job-shop上进行处理。对于每一个批量，它对设备的工况要求是不同的，但是在加工特定批量时，设备的工况是固定的。在一个定周期预防性维护(PM)周期内，设备加工不同的批量导致的设备的可靠性变化如图1所示。图1中，虚线表示在基准工况下设备的可靠性，红线和绿线分别表示在不同批量下设备的可靠性变化。Suppose a job-shop contains M devices. A processing task including N batches needs to be processed on this job-shop. For each batch, it has different requirements for the working conditions of the equipment, but when processing a specific batch, the working conditions of the equipment are fixed. In a regular preventive maintenance (PM) cycle, the reliability changes of the equipment caused by different batches of equipment processing are shown in Figure 1. In Figure 1, the dotted line represents the reliability of the equipment under the benchmark conditions, and the red line and the green line represent the reliability changes of the equipment under different batches respectively.

在加工任务开始前所有的批量都已准备就绪。这些批量之间是相互独立的。对于每一个批量都有一个交货截止时间，如果批量的完成时间滞后于相应的交货时间，则会产生延迟成本作为成本。设备为转换不同批量的启动时间可以忽略不计。另外，批量间的转移时间是可以忽略的。All batches are ready before the machining job starts. These batches are independent of each other. There is a delivery cut-off time for each batch, and if the completion time of the batch lags behind the corresponding delivery time, delay costs will be incurred as costs. The start-up time of the equipment for converting different batches is negligible. In addition, the transfer time between batches is negligible.

如果设备发生故障，则对设备进行小修，使其恢复到可以运行的状态但是不改变其风险率。为了简化计算，设备的小修时间忽略不计。假设设备上进行成批生产模式，但是预防性维护PM不能打断一个正在进行加工的设备。维护不得不提前至加工任务前，或推迟至加工任务后，如图2所示。If the equipment fails, minor repairs are made to the equipment to restore it to an operable state without changing its risk rate. In order to simplify the calculation, the minor repair time of the equipment is ignored. It is assumed that batch production mode is performed on the equipment, but the preventive maintenance PM cannot interrupt an equipment that is being processed. Maintenance has to be advanced before the processing task, or postponed until after the processing task, as shown in Figure 2.

本发明中的基本假设和符号说明见下表1：Basic assumptions and symbolic descriptions in the present invention are shown in the following table 1:

表1：设备机会维护和生产调度集成优化模型数据说明Table 1: Data description of the integrated optimization model of equipment opportunity maintenance and production scheduling

A1：加工任务伊始，所有的设备都是新的。A1: At the beginning of the processing task, all the equipment is new.

A2：如果设备发生了故障，对设备进行小修并且小修仅能使设备“as bad asold”。预防性维护PM是非完美性的，能够使设备恢复到一个较好但非新的状态。A2: If the equipment fails, make minor repairs to the equipment and minor repairs can only make the equipment "as bad asold". Preventive maintenance (PM) is imperfect and restores equipment to a better but not new condition.

A3：PM必须是在批量加工前或批量加工后进行。A3: PM must be carried out before or after batch processing.

A4：小修时间、设备启动时间、批量转换和运输时间是可以忽略的。A4: Minor repair time, equipment start-up time, batch conversion and shipping time are negligible.

A5：在加工任务开始时，所有的批量准备就绪，取消一个正在加工的批量而加工其他批量是不允许的。A5: At the beginning of the processing task, all batches are ready, it is not allowed to cancel a batch being processed while processing other batches.

A6：不同的批量要求的工况也不一样，但是在一个批量下，工况是不变的。A6: Different batches require different working conditions, but in one batch, the working conditions remain the same.

A7：在一个时刻设备仅能被一个批量占用，且一个批量仅在一台设备上加工。A7: The equipment can only be occupied by one batch at a time, and one batch can only be processed on one equipment.

Figure GDA0003719825160000091
Figure GDA0003719825160000091

Figure GDA0003719825160000101
Figure GDA0003719825160000101

表1Table 1

本发明方法部分2：确定优化目标Part 2 of the method of the present invention: Determining the optimization target

本发明的集成优化是指通过确定job-shop上各个设备上的加工顺序和PM的执行时间使总成本最小。总成本包括延迟成本、PM成本和期望小修成本。一个批量的延迟成本与它的完成时间有关。PM成本和小修成本与PM的执行次数和时间有关。本文的目标函数可以写成：The integrated optimization of the present invention refers to minimizing the total cost by determining the processing sequence on each device on the job-shop and the execution time of PM. The total cost includes delay cost, PM cost and expected minor repair cost. The delay cost of a batch is related to its completion time. PM cost and minor repair cost are related to the number and time of PM execution. The objective function of this paper can be written as:

min Ctotal＝CT+CP+CF 公式1min Ctotal ＝CT +CP +CF Formula 1

式中，Ctotal为完成全部生产任务后的总成本，CT为总延迟成本，CP为总PM成本，CF为总期望小修成本。In the formula, Ctotal is the total cost after completing all production tasks, CT is the total delay cost, CP is the total PM cost, and CF is the total expected minor repair cost.

Figure GDA0003719825160000102
Figure GDA0003719825160000102

式中，N为批量总数，Tn为批量n的延迟成本。

Figure GDA0003719825160000103
为批量n的单位时间内延迟成本。In the formula, N is the total number of batches, and Tn is the delay cost of batch n.
Figure GDA0003719825160000103
Delay cost per unit time for batch n.

Figure GDA0003719825160000104
Figure GDA0003719825160000104

其中，Ln为批量n的延迟。Among them, Ln is the delay of batch n.

Figure GDA0003719825160000111
Figure GDA0003719825160000111

式中，Em,n为批量n在设备m上的完成时间。Dn为给定的批量n的预计完成时间In the formula, Em,n is the completion time of batch n on equipment m. Dn is the expected completion time of a given batch n

Figure GDA0003719825160000112
Figure GDA0003719825160000112

式中，Im为任务周期内设备m上执行预防性维护PM的总次数，

Figure GDA0003719825160000113
为一次预防性维护PM成本。In the formula, Im is the total number of PM executions of preventive maintenance on device m in the task period,
Figure GDA0003719825160000113
PM cost for one preventive maintenance.

Figure GDA0003719825160000114
Figure GDA0003719825160000114

式中，

Figure GDA0003719825160000115
为在基准工况下设备m在第i个预防性维护PM周期内的等价加工时间，
Figure GDA0003719825160000116
为在基准工况下设备m进行完最后一次预防性维护PM后的等价加工时间，
Figure GDA0003719825160000117
为第i个预防性维护PM周期内设备m的基准失效率函数。In the formula,
Figure GDA0003719825160000115
is the equivalent processing time of equipment m in the i-th preventive maintenance PM cycle under the baseline condition,
Figure GDA0003719825160000116
is the equivalent processing time after the last preventive maintenance PM of equipment m under the baseline condition,
Figure GDA0003719825160000117
is the baseline failure rate function of equipment m in the i-th preventive maintenance PM cycle.

本发明方法部分3：确定批量完成时间Inventive method part 3: Determining batch completion time

一台设备上一个批量的完成时间是很复杂的。它主要与四项因素有关：(1)此台设备上上一批量的完成时间；(2)此台设备上此批量的加工时间；(3)此批量在上一台设备上的完成时间；(4)如果在加工此批量前需要进行PM，则PM的执行时间。The completion time of a batch on a device is complex. It is mainly related to four factors: (1) the completion time of the previous batch on this equipment; (2) the processing time of this batch on this equipment; (3) the completion time of this batch on the previous equipment; (4) If PM is required before processing this batch, the execution time of PM.

在每个设备上的第一个加工的完成时间是计算在此job-shop中其他批量的完成时间的基础。需要考虑两种情景。一是在此job-shop上分配给每台设备上的第一个加工批量互不相同。另一个情景是在此job-shop上分配给每台设备上的第一个加工批量有相同的。对于第一种情景，则第一个加工的完成时间为：The completion time of the first process on each machine is the basis for calculating the completion time of other batches in this job-shop. Two scenarios need to be considered. One is that the first processing batch assigned to each device in this job-shop is different from each other. Another scenario is that the first processing batch assigned to each device on this job-shop has the same. For the first scenario, the completion time of the first process is:

Figure GDA0003719825160000118
Figure GDA0003719825160000118

其中，Em,n,1为在设备m上如果第1个加工是批量n它的完成时间；xm,n,1为0-1变量，如果设备m上第1个加工是批量n，则xm,n,1的值为1；pm,n为批量n在设备m上的加工时间。公式7的约束说明在一个时刻，设备只能被一个批量占用且一个批量仅能占用一个设备。Among them, Em,n,1 is the completion time if the first processing on equipment m is batch n; xm,n,1 is a 0-1 variable, if the first processing on equipment m is batch n, Then the value of xm,n,1 is 1; pm,n is the processing time of batch n on equipment m. The constraint of formula 7 shows that at a moment, a device can only be occupied by one batch and a batch can only occupy one device.

对于第二种情景，在一个设备上的第1个加工完成时间和该批量在其他设备上的完成时间有关，如公式8所示。For the second scenario, the completion time of the first processing on one device is related to the completion time of the batch on other devices, as shown in Equation 8.

Figure GDA0003719825160000121
Figure GDA0003719825160000121

如果在设备m上的第k次加工为批量n，它的完成时间能从以下几个方面进行推断：如果该批量已经在其他设备上进行了加工，则它的完成时间；设备m上第k-1个加工的完成时间；如果要执行PM，则PM的执行时间。其计算公式如公式9所示：If the kth processing on equipment m is batch n, its completion time can be inferred from the following aspects: if the batch has been processed on other equipment, its completion time; - Completion time of 1 machining; if PM is to be performed, execution time of PM. Its calculation formula is shown in formula 9:

Figure GDA0003719825160000122
Figure GDA0003719825160000122

其中，

Figure GDA0003719825160000123
为设备m上的执行一次预防性维护PM所需的时间，k与k'相等或不相等，如果k＝k'，说明在设备m和设备m'上批量n所处的顺序是相同的，否则，设备m和设备m'上批量n所处的顺序不相同。in,
Figure GDA0003719825160000123
It is the time required to perform a preventive maintenance PM on equipment m, k and k' are equal or not equal, if k=k', it means that the order of batch n on equipment m and equipment m' is the same, Otherwise, the order of batch n on device m and device m' is not the same.

本发明方法部分4：确定PM周期内的设备可靠性Inventive Method Part 4: Determining Equipment Reliability During a PM Period

执行一次PM的初始时间是根据在设备上最小维护成本率确定的。一台正在加工的设备不能因为执行PM被中断，所以PM不得不被提前或推后至加工任务完成。确定初始执行PM时间的成本率函数为公式10The initial time to execute a PM is determined according to the minimum maintenance cost rate on the device. A machine that is being processed cannot be interrupted by executing PM, so PM has to be advanced or postponed until the processing task is completed. The cost rate function to determine the initial execution PM time is Equation 10

Figure GDA0003719825160000124
Figure GDA0003719825160000124

其中，

Figure GDA0003719825160000125
为设备m在第i个预防性维护PM周期内的维护成本率，
Figure GDA0003719825160000126
为设备m上的执行一次预防性维护PM所需的时间。in,
Figure GDA0003719825160000125
is the maintenance cost rate of equipment m in the i-th PM cycle of preventive maintenance,
Figure GDA0003719825160000126
is the time required to perform a preventive maintenance PM on device m.

由于设备工作在不同的工况下，设备风险率也随着加工批量的不同而不同。假设在设备m的第i个周期内已经有

Figure GDA0003719825160000131
个加工完或正在加工的批量，在该周期内设备的风险率函数写为：Since the equipment works under different working conditions, the equipment risk rate also varies with the processing batch. Assume that during the i-th cycle of device m there has been
Figure GDA0003719825160000131
A batch that has been processed or is being processed, the risk rate function of the equipment in this period is written as:

Figure GDA0003719825160000132
Figure GDA0003719825160000132

式中，

Figure GDA0003719825160000133
是0-1变量，如果设备m上的第i个预防性维护PM周期内的第j个加工是批量n，则为1；
Figure GDA0003719825160000134
为在第i个预防性维护PM周期中第j个加工过程中设备m的风险率函数。In the formula,
Figure GDA0003719825160000133
is a 0-1 variable, which is 1 if the j-th process in the i-th preventive maintenance PM cycle on equipment m is batch n;
Figure GDA0003719825160000134
is the risk rate function of equipment m in the j-th process in the i-th preventive maintenance PM cycle.

时变的工况可以通过AFTM转换成基准工况。以第1个PM周期为例，在第1个PM周期内设备在进行第1项加工时的可靠性为：Time-varying operating conditions can be converted to baseline operating conditions by AFTM. Taking the first PM cycle as an example, the reliability of the equipment during the first processing in the first PM cycle is:

Figure GDA0003719825160000135
Figure GDA0003719825160000135

其中，

Figure GDA0003719825160000136
表示在第1个预防性维护PM周期内设备在进行第1项加工时的可靠性，
Figure GDA0003719825160000137
是在基准工况下设备的等价可靠性，
Figure GDA0003719825160000138
是设备m上第k个加工时的工况调节参数，
Figure GDA0003719825160000139
为在基准工况下在第i个预防性维护PM周期内设备m的风险率函数。in,
Figure GDA0003719825160000136
Indicates the reliability of the equipment during the first item of processing during the first PM cycle of preventive maintenance,
Figure GDA0003719825160000137
is the equivalent reliability of the equipment under the reference condition,
Figure GDA0003719825160000138
is the working condition adjustment parameter of the kth processing on the equipment m,
Figure GDA0003719825160000139
is the risk rate function of equipment m in the ith preventive maintenance PM cycle under the baseline condition.

则，在第1个PM周期内设备m在进行第1个加工是设备的风险率函数为：Then, the risk rate function of equipment m performing the first processing in the first PM cycle is:

Figure GDA00037198251600001310
Figure GDA00037198251600001310

在第1个PM周期内，设备m在进行第2个加工时设备的可靠性为：In the first PM cycle, the reliability of the equipment m during the second processing is:

Figure GDA0003719825160000141
Figure GDA0003719825160000141

其中，in,

Figure GDA0003719825160000142
Figure GDA0003719825160000142

其中，Sm,n,k为如果设备m上第k个加工是批量n，设备m的加工开始时间。设备m在第1个PM周期内第2个加工时设备的风险率函数为：Among them, Sm,n,k is the processing start time of equipment m if the k-th processing on equipment m is batch n. The risk rate function of equipment m during the second processing in the first PM cycle is:

Figure GDA0003719825160000143
Figure GDA0003719825160000143

设备m在第1个PM周期内第k次加工时设备m的可靠性为：The reliability of equipment m during the kth processing of equipment m in the first PM cycle is:

Figure GDA0003719825160000144
Figure GDA0003719825160000144

设备m在第1个PM周期内第k次加工过程中设备的风险率函数为：The risk rate function of equipment m during the kth processing in the first PM cycle is:

Figure GDA0003719825160000151
Figure GDA0003719825160000151

由于前文假设，PM为非完美性，引入役龄递减因子和失效率递增因子。设备m在第i个PM周期内设备的基准风险率函数为：Due to the previous assumption that PM is not perfect, the service age decreasing factor and the failure rate increasing factor are introduced. The basic risk rate function of equipment m in the i-th PM period is:

Figure GDA0003719825160000152
Figure GDA0003719825160000152

本发明方法部分5：使用随机键遗传算法GA进行集成优化Part 5 of the method of the present invention: Integrated optimization using the random key genetic algorithm GA

首先，定义染色体类型。考虑到此为多设备多批量问题，因此引入染色体组的概念。染色体组是由两部分构成，第1部分为设备上批量的分配，它是一个排序问题，因此，产生一些随机键并将它们按照大小排序，进而得到了加工顺序；第2部分为PM决策，此不为排序问题，因此没有必要对它们分配随机键。First, define the chromosome type. Considering that this is a multi-device and multi-batch problem, the concept of chromosome set is introduced. The genome is composed of two parts. The first part is the batch allocation on the device. It is a sorting problem. Therefore, some random keys are generated and sorted according to their size, and then the processing order is obtained; the second part is PM decision, This is not a matter of ordering, so there is no need to assign random keys to them.

随后，进行交叉。计算经过编码后每个染色体组的适应度，以本方法的目标函数作为适应度函数。为保持GA的进化为单调非减性，采用精英策略。在每次迭代过程中把精英记录下来并放回交配池中。仅在加工顺序部分执行交叉操作。因为PM的机会是根据最小维修成本率获得的，所以在PM决策部分不进行交叉操作。最小维修成本率与设备在执行前一个PM之后的服役时间有关。也就是说，与在PM之后的加工顺序有关。因此PM决策矩阵是与加工顺序部分有关。交叉操作仅作用于随机键部分。所有的染色体组和他们的键分为两部分：一部分为精英组，一部分为非精英组。精英组中的染色体组和非精英组中的染色体组进行交配。执行单点交叉，并进入凸集理论加速算法的收敛。在交叉过程中，交叉点随机选择。父代在交叉点左侧保持不变，在右侧的部分彼此交换。使用线性连接方法获得交叉点的数值。Subsequently, a crossover is performed. Calculate the fitness of each chromosome group after encoding, and use the objective function of this method as the fitness function. In order to keep the evolution of GA as monotonous and non-subtractive, an elite strategy is adopted. Elites were recorded and returned to the mating pool during each iteration. Interleave operations are performed only in part of the machining sequence. Because PM opportunities are obtained based on the minimum maintenance cost rate, no crossover is performed in the PM decision part. The minimum repair cost rate is related to how long the equipment has been in service since the previous PM was performed. That is, it is related to the processing sequence after PM. Therefore the PM decision matrix is partly related to the processing sequence. The crossover operation only works on the random key part. All chromosome sets and their bonds are divided into two parts: an elite group and a non-elite group. The sets of chromosomes in the elite group are mated with the sets of chromosomes in the non-elite group. Performing one-point crossover and entering convex set theory speeds up the convergence of the algorithm. During the crossover process, the intersection point is chosen randomly. The parents remain unchanged on the left side of the intersection, and the parts on the right side are swapped with each other. Use the linear join method to obtain the value of the intersection point.

之后，进行变异。变异可以发生在加工顺序部分和PM决策部分。在一个染色体上随机的选择两个基因，然后执行凸连接变异。与交叉算子相同，变异也仅在随机键上进行。对于第二种情况，突变在PM决策部分进行。在PM决策部分，选择某个非零的位置，因为非零的位置为理论上执行PM的时间。在突变位置，突变后的结果为1减去原来的值。After that, mutate. Variation can occur in the processing sequence part and the PM decision part. Randomly select two genes on a chromosome and perform convex join mutation. Like the crossover operator, mutations are also performed only on random keys. For the second case, mutations are made in the PM decision part. In the PM decision part, choose some non-zero position, because the non-zero position is the theoretical time to perform PM. At the mutated position, the mutated result is 1 minus the original value.

需要说明的是，如果在PM决策部分发生了突变，则在设备上的加工顺序保持不变，在突变发生后，重新计算适应度。重复选择、交叉、突变过程，直到当前迭代次数超过最大迭代次数的限制。It should be noted that if a mutation occurs in the PM decision-making part, the processing sequence on the equipment remains unchanged, and the fitness is recalculated after the mutation occurs. Repeat the process of selection, crossover and mutation until the current number of iterations exceeds the limit of the maximum number of iterations.

Claims (3)

1. A method for integrating and optimizing equipment opportunity maintenance and production scheduling under a time-varying working condition is characterized by comprising the following steps:
first step, determining problems and assumptions
Assuming that a job contains M devices, a processing task comprising N batches is to be processed on the job, all the batches are ready before the processing task is started, and the N batches are independent; there is a delivery deadline for each lot, and if the completion time of the lot lags behind the corresponding delivery deadline, a delay cost is incurred as a cost; the starting time for converting different batches and the transfer time between batches are ignored; if the equipment fails, performing minor repair on the equipment to restore the equipment to a state capable of running without changing the risk rate of the equipment, and neglecting the minor repair time of the equipment; assuming a batch mode is in progress on the equipment, but the preventative maintenance PM cannot interrupt an equipment that is in process, so the preventative maintenance PM has to be advanced to before or deferred to after the process job;
second, determining an optimization objective
The integration optimization means that the total cost is minimized by determining the processing sequence of each device on the job-shop and the execution time of the preventive maintenance PM, wherein the total cost comprises delay cost, preventive maintenance PM cost and expected minor repair cost, the delay cost of one batch is related to the completion time of the batch, the preventive maintenance PM cost and the minor repair cost are related to the execution times and time of the preventive maintenance PM, and the objective function is as follows:
min Ctotal ＝CT +CP +CF
in the formula, Ctotal Is the total cost;
CT in order to account for the total delay cost,
Figure FDA0003719825150000011
n is the total number of batches, Tn For the delay cost of the batch n,
Figure FDA0003719825150000012
Ln is a delay of the batch n and,
Figure FDA0003719825150000013
Em,n for the completion time of batch n on device m, Dn For a given projected completion time of batch n,
Figure FDA0003719825150000014
delay cost per unit time for batch n;
CP PM cost for total preventative maintenance:
Figure FDA0003719825150000015
Im for the total number of times the preventive maintenance PM is performed on the device m during the mission period,
Figure FDA0003719825150000021
cost of PM for one-time preventive maintenance;
CF for the total desired minor repair cost:
Figure FDA0003719825150000022
Figure FDA0003719825150000023
the cost of the previous minor repair of the equipment m,
Figure FDA0003719825150000024
for an equivalent machining time of the machine m in the ith preventive maintenance PM cycle at the base condition,
Figure FDA0003719825150000025
in order to achieve equivalent processing time after the equipment m performs the last preventive maintenance PM under the reference working condition,
Figure FDA0003719825150000026
a reference failure rate function of the device m in the ith preventive maintenance PM period;
thirdly, determining the batch completion time
The completion time of a batch on a piece of equipment is related to four factors: (1) the completion time of the last batch on the equipment; (2) the processing time of the batch on the equipment; (3) the completion time of the batch on the previous equipment; (4) An execution time for preventively maintaining the PM if the preventive maintenance PM is required before the lot is processed;
step four, determining the equipment reliability in the PM period of preventive maintenance
The cost rate function for determining the time to initially perform preventative maintenance PM is given by:
Figure FDA0003719825150000027
Figure FDA0003719825150000028
in the formula,
Figure FDA0003719825150000029
the maintenance cost rate for the device m during the ith preventative maintenance PM cycle,
Figure FDA00037198251500000210
the time required to perform a preventive maintenance PM for the equipment m;
suppose there is already an i-th cycle of device m
Figure FDA00037198251500000211
For a finished or in-process lot, the risk rate function for the equipment during that cycle is written as:
Figure FDA0003719825150000031
Figure FDA0003719825150000032
Figure FDA0003719825150000033
in the formula,
Figure FDA0003719825150000034
is a variable from 0 to 1, 1 if the jth process in the ith preventative maintenance PM cycle on device m is lot n;
Figure FDA0003719825150000035
as a function of the risk of equipment m in the jth process in the ith preventive maintenance PM cycle;
the reliability of the equipment in the 1 st preventive maintenance PM period when the equipment carries out the 1 st processing is as follows:
Figure FDA0003719825150000036
Figure FDA0003719825150000037
in the formula,
Figure FDA0003719825150000038
indicating the reliability of the equipment at the time of the 1 st process in the 1 st preventive maintenance PM cycle,
Figure FDA0003719825150000039
is the equivalent reliability of the device under the baseline conditions,
Figure FDA00037198251500000310
is a working condition adjusting parameter in the k-th processing on the equipment m,
Figure FDA00037198251500000311
is a risk rate function of the equipment m in the 1 st preventive maintenance PM period under the reference condition;
the risk rate function of the equipment m during the 1 st process in the 1 st preventive maintenance PM cycle is:
Figure FDA00037198251500000312
Figure FDA00037198251500000313
in the 1 st preventive maintenance PM period, the reliability of the equipment m when performing the 2 nd processing is as follows:
Figure FDA0003719825150000041
Figure FDA0003719825150000042
Figure FDA0003719825150000043
wherein:
Figure FDA0003719825150000044
Figure FDA0003719825150000045
Figure FDA0003719825150000046
wherein S ism,n,k The processing start time of the equipment m if the kth processing on the equipment m is the batch n; the risk rate function of the equipment m at the 2 nd process in the 1 st preventative maintenance PM cycle is:
Figure FDA0003719825150000047
Figure FDA0003719825150000048
Figure FDA0003719825150000049
the reliability of the apparatus m at the kth processing in the 1 st preventive maintenance PM cycle is:
Figure FDA00037198251500000410
Figure FDA00037198251500000411
Figure FDA00037198251500000412
the risk rate function for the equipment m during the kth process in the 1 st preventive maintenance PM cycle is:
Figure FDA0003719825150000051
Figure FDA0003719825150000052
m＝1,2,…,M n＝1,2,…,N n'＝1,2,…,N
preventive maintenance PM is non-perfect, introducing a work-age decreasing factor and a failure rate increasing factor, and the benchmark risk rate function for equipment m during the ith preventive maintenance PM cycle is:
Figure FDA0003719825150000053
and fifthly, performing integrated optimization by using a random key genetic algorithm GA.
2. The method for integrated optimization of equipment opportunity maintenance and production scheduling under the time-varying working condition of claim 1, wherein the completion time of other batches in the job is calculated by calculating the completion time of the first process on each equipment, and the following steps are carried out:
if the first processing batches allocated to each equipment in the job-shop are different from each other, the completion time of the first processing on the equipment is as follows:
Figure FDA0003719825150000054
Figure FDA0003719825150000055
in the formula, Em,n,1 If the 1 st process is a batch n, its completion time on equipment m; x is the number ofm,n,1 Is a variable from 0 to 1, x if the 1 st process on plant m is a batch nm,n,1 Has a value of 1; p is a radical of formulam,n Is the processing time of batch n on the equipment m;
if the first processing batch allocated to each equipment on the job-shop is the same, the completion time of the first processing on the equipment is as follows:
Figure FDA0003719825150000061
Figure FDA0003719825150000062
Figure FDA0003719825150000063
if the kth processing on machine m is a batch n, its completion time is inferred from the following: if the batch has been processed on other equipment, its completion time; finish time of the (k-1) th process on the equipment m; if the preventive maintenance PM is to be executed, the execution time of the preventive maintenance PM is calculated by the following formula:
Figure FDA0003719825150000064
Figure FDA0003719825150000065
Figure FDA0003719825150000066
Figure FDA0003719825150000067
Figure FDA0003719825150000068
in the formula,
Figure FDA0003719825150000069
the time required to perform a preventive maintenance PM on device m, k being equal or not equal to k ', if k = k', it means that the order in which the batch n is placed on device m and device m 'is the same, otherwise, the order in which the batch n is placed on device m and device m' is not the same.
3. The time-varying operating condition equipment opportunity maintenance and production scheduling integrated optimization method of claim 1, wherein the fifth step comprises the steps of:
firstly, defining chromosome type, wherein a chromosome group is composed of two parts, the part 1 is the batch distribution on equipment and is a sequencing problem, therefore, random keys are generated and are sequenced according to sizes, and further, the processing sequence is obtained; part 2 is a preventative maintenance PM decision, which is not a ranking problem;
subsequently, a crossover is performed: calculating the fitness of each chromosome set after coding, taking a target function as a fitness function, adopting an elite strategy to keep the evolution of a random key genetic algorithm GA to be monotonous and non-decreasing, recording elite in each iteration process, putting the elite back into a mating pool, and only performing cross operation on a processing sequence part; the cross operation is not carried out in a PM preventive maintenance decision part, the minimum maintenance cost rate is related to the service time of the equipment after the execution of the previous PM preventive maintenance, a PM preventive maintenance decision matrix is related to a processing sequence part, and the cross operation only acts on a random key part; all the chromosomes and their bonds are divided into two parts: one part is an elite group, the other part is a non-elite group, and a chromosome group in the elite group and a chromosome group in the non-elite group are mated, a single-point crossing is carried out, and convergence of a convex set theory acceleration algorithm is entered; in the crossing process, the crossing points are randomly selected, the parents keep unchanged at the left sides of the crossing points, the parts at the right sides are exchanged with each other, and the numerical values of the crossing points are obtained by using a linear connection method;
then, mutation was performed: mutation occurs in a processing sequence part and a preventive maintenance PM decision part, two genes are randomly selected on a chromosome, convex connection mutation is then executed, mutation is only carried out on random keys as with a crossover operator, mutation is carried out in the preventive maintenance PM decision part for the case that mutation occurs in the preventive maintenance PM decision part, and a certain non-zero position is selected in the preventive maintenance PM decision part because the non-zero position is the time for theoretically executing preventive maintenance PM; at the mutation site, the result after mutation was 1 minus the original value.
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